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Social
Social Science Research xxx (2005) xxx–xxx
Science
RESEARCH
www.elsevier.com/locate/ssresearch
Racial similarity in the relationship
between poverty and homicide rates:
Comparing retransformed coefficients
Lance Hannon a,*, Peter Knapp b, Robert DeFina b
a
Department of Sociology, Occidental College, 1600 Campus Road, Los Angeles, CA 90041, USA
b
Villanova University, USA
Abstract
Criminologists have shown great interest in comparing the strength of the relationship
between poverty and violent crime for whites and blacks. The present paper argues that the standard approach of comparing race-specific coefficients from logarithmic metric OLS and/or Poisson-based regressions has led to erroneous conclusions in this literature. Unlike researchers in
other disciplines (especially economics), criminologists have largely ignored the need to ‘‘retransform’’ coefficients to their linear-effect representations before making comparisons
between groups. The current study illustrates the importance of this methodological issue for
the substantive question of whether povertyÕs relationship to homicide is racially invariant
(N = 134 cities). Similar to previous studies, initial results indicated that povertyÕs effect on
the natural logarithm of the homicide rate was dramatically stronger for whites than blacks
(nearly 300% stronger for whites). However, after applying a broadly useful retransformation
formula, povertyÕs effect on the homicide rate actually appeared somewhat stronger for blacks.
Further application of bootstrap simulations necessary to calculate the standard error of the difference in coefficients suggested that this racial discrepancy was not statistically significant.
Ó 2005 Elsevier Inc. All rights reserved.
Keywords: Homicide; Race-specific; Poverty; Disadvantage; Poisson; Logarithmic
*
Corresponding author.
E-mail address: [email protected] (L. Hannon).
0049-089X/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.ssresearch.2005.01.004
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1. Introduction
As Kubrin and Wadsworth (2003, p. 3) have pointed out, ‘‘one of the most
intriguing findings’’ reported in race-specific studies is that important structural factors like poverty appear to have less of an impact on black homicide than they do on
white homicide. Similarly, Messner and Rosenfeld (1999, p. 37) have argued
Despite extensive work on the social-structural sources of homicide during
recent decades, important tasks need to be confronted. One such task is to
explain the more curious anomalies in the literature. As noted above, poverty
(and the correlates of poverty) emerge as major determinants of homicide
offending for Whites but not for Blacks. Researchers have identified other factors that help explain levels of Black offending, such as residential segregation,
but the question still remains why the effects of poverty vary so dramatically
across racial groups.
The answer to this question may have important implications for social disorganization theory and the black subculture of violence thesis. Social disorganization theory
generally hypothesizes racial invariance in the detrimental effects of poverty (Sampson and Wilson, 1995). According to this view, the structural conditions associated
with neighborhood poverty universally inhibit collective efficacy and the ability of a
community to police itself. In contrast, the common claim that poverty has a substantially weaker effect on black homicide rates can be interpreted as supportive
of certain versions of the black subculture of violence perspective, particularly those
that suggest that subcultural socialization can moderate the usual influence of structural factors on human behavior (Messner, 1983).
In a well-cited study that laid out both the theoretical and policy significance of
this topic, Ousey (1999) demonstrated that various indicators of economic deprivation have weaker effects on the black logarithmically transformed homicide rate. Ousey (1999, pp. 416–417) noted
. . .While the association between poverty and homicide is statistically significant in the regression models for both racial groups, the size of the association
varies by race. A close comparison of the poverty coefficients in the two models
reveals that the coefficient is nearly four times greater in the model for whites. . .
In fact, the results suggest that all else being equal, white homicide rates would
outdistance black homicide rates if poverty rates were equivalent.
Similarly, in an important analysis of the relationship between race-specific concentrated disadvantage and race-specific logarithmically transformed homicide rates,
Krivo and Peterson (2000, p. 552) concluded
Concentrated disadvantage has a significant positive influence for both whites
and blacks, but the effect for whites is nearly twice as large as that for blacks.
Krivo and Peterson suggested that this discrepancy may be a byproduct of nonlinearity in the community disadvantage/homicide relationship, while Ousey (1999)
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argued that his findings demonstrate the need to revise traditional structural theories
to include the notion that subcultural adaptations to deprivation may be a unique
driving force behind violence in black communities.
Besides their contributions in terms of framing the relevant theoretical and policy
issues, we argue that these and several other recent cross-sectional studies on racespecific rates of violence (e.g. Lee, 2000; McNulty, 2001; Phillips, 2002; Wooldredge
and Thistlethwaite, 2003) have gone substantially beyond previous analyses by correctly insisting on comparing race-specific effects, not just associated probability values. However, most recent analyses have unfortunately followed a common practice
in earlier studies of ignoring data transformations in the comparison of race-specific
results. Specifically, previous studies have generally employed Poisson or semi-log
regression models where the dependent variable is log transformed and have assumed that showing racially varying effects on the logarithm of the homicide rate
is equivalent to demonstrating racially varying effects on the homicide rate, the outcome variable with which the general public and policy makers are most familiar.
While other disciplines (especially economics, engineering, and public health)
have developed substantial literatures around the interpretation of transformed
variables in terms of their relevance for the original theoretically meaningful metrics, criminologists have often ignored issues of variable ‘‘retransformation,’’ resulting in methodological approaches that are sometimes incongruent with their
theoretical concerns. Hannon and Knapp (2003) noted this mismatch between
method and theory in their assessment of the use of variable transformations in
studies on non-linearity in the disadvantage/violent crime relationship. Their analysis suggested that some previously reported results in this area were statistical artifacts of log transformation of the dependent variable. Hannon and Knapp
concluded their paper arguing that logarithmic transformation, or implicit logarithmic transformation as in the case of the popular negative binomial model, has
unfortunately become a misunderstood tradition in a wide range of criminological
and sociological studies.
Addressing a now extensive collection of homicide studies focusing on the comparison of race-specific coefficients, the present paper generalizes the cautionary tale
from Hannon and Knapp (2003) about how variable transformations obstruct the
assessment of non-linearity to how such transformations can also lead to erroneous
inferences regarding differences between groups. Moreover, the present paper offers
solutions to the statistical retransformation problems that Hannon and KnappÕs
analysis raises, but does not solve. These solutions include a broadly applicable
and precise formula for retransforming coefficients and a procedure for computing
standard errors for retransformed estimates. Using race-specific city-level homicide
offending data and emphasizing their relation to race-specific poverty rates, the current study illustrates how the misuse of variable transformations can be a serious
obstacle to answering the substantive question of whether economic deprivation
has a stronger effect on white than black homicide rates. In addition, the present paper provides a way for researchers in this area (and other areas) to reap the methodological benefits of variable transformations without suffering the loss in terms of
substantive interpretability.
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2. Comparing effects across samples in race-specific homicide studies: itÕs the
significance level of the difference that matters
There is now a substantial literature focused on the question of whether various
theoretically relevant predictors of homicide are invariant in their importance for
blacks and whites (Chilton and Regoeczi, 2004; Hannon and DeFina, 2005; Hawkins, 1999; Krivo and Peterson, 1999; LaFree and Drass, 1996; Messner and Golden,
1992; Messner and Rosenfeld, 1999; Parker and Pruitt, 2000a,b; Rose and McClain,
1990; Shihadeh and Ousey, 1996). There are several excellent reviews of this literature (Messner et al., 2001; Parker and McCall, 1999; Phillips, 2002; Wooldredge
and Thistlethwaite, 2003). In terms of the invariance of the effect of economic disadvantage on homicide, previous studies have sometimes been inconsistent. While most
studies conclude that poverty-related factors have a significantly weaker effect on
black rates than white rates (Harer and Steffensmeier, 1992; Huff-Corzine et al.,
1986; Krivo and Peterson, 2000; Ousey, 1999; Smith, 1992), a few conclude that poverty-related factors are similar determinants for blacks and whites (Lee, 2000; Morenoff et al., 2001; Sampson, 1985, 1987), and one even concludes that poverty-related
factors have a somewhat weaker impact on white rates than black rates (Phillips,
2002). This diversity of conclusions is likely a reflection of the complexity of the theoretical issues surrounding this topic, as well as the complexity of methods used to
assess them.
OuseyÕs (1999) comprehensive analysis of the structural covariates of urban racespecific homicide rates provides a useful framework for critically assessing and building on previous research. Specifically, OuseyÕs (1999) refinements in the procedure
for testing the racial invariance hypothesis call attention to several important limitations of most previous studies. Foremost, rather than inferring significant differences
between coefficients in the white and black models from individual probability values, as almost all previous studies had done, Ousey argued for the importance of formally testing for significant differences in effects. Previous studies had implicitly
assumed that finding a significant effect for whites and not for blacks means that
the effect is significantly stronger for whites.1 This is problematic because what really
matters for this conclusion is the significance of the difference in estimates, which can
only be determined by taking into account the joint confidence interval. As Matthews and Altman (1996, p. 808) state in their assessment of the most common errors
in statistical interaction studies,
There is a temptation to claim that the difference in P values establishes a difference between subgroups. . . This argument is false. . .A P value is a composite
1
Of the dozen studies Ousey reviewed, only Sampson (1987) explicitly tested for significant differences
between estimates. We believe that OuseyÕs (1999) analysis, in conjunction with Paternoster et al.Õs (1998)
methodological paper, has helped to create a situation where most studies now properly compare racespecific effects rather than just the results of separate tests for statistical significance. However, we are
aware of a few prominent recent studies where racial differences in effects are still inferred from racial
differences in probability values. Thus, we believe the point needs to be stressed.
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which depends not only on the size of an effect but also on how precisely the
effect has been estimated (its standard error). So differences in P values can
arise because of differences in effect sizes or differences in standard errors or
a combination of the two.
Thus, a simple comparison of individual probability values ignores the crucial distinction between the size of an estimated effect and how precisely it has been estimated2 (also see Sampson, 1987). Indeed, it is relatively common when comparing
effects from two different samples to find only one coefficient statistically significant,
and yet, taking into account the degree of overlap in the two confidence intervals for
the estimates, the difference between the two effects is statistically insignificant3 (see
Paternoster et al., 1998). Ousey (1999) addressed this issue by explicitly testing for
significant racial differences in the estimated effects of indicators of structural
disadvantage.
Other procedural refinements recommended by Ousey include the use of relatively
parsimonious models with close attention to maintaining moderate (and roughly
equal) levels of multicollinearity, and the use of seemingly unrelated regression to account for correlated disturbance terms in non-independent samples. Following these
guidelines, Ousey analyzed the separate effects of race-specific measures of the poverty rate, the unemployment rate, income inequality, the prevalence of femaleheaded families, and a general deprivation index on logarithmically transformed
race-specific homicide offending rates for large cities (N = 125). He reported that,
with the exception of income equality, there is clear evidence of weaker effects for
these measures of structural disadvantage for blacks than whites. For example, the
effect of poverty on the natural logarithm of the homicide rate for blacks
(b = .023, SE = .008) was significantly less than that for whites (b = .086,
SE = .013); p < .05 for the difference. Ousey concluded that these pronounced racial
differences in the effects of structural factors suggest the need to incorporate cultural
factors to explain the racial gap in homicide offending.
Three other recent homicide studies should also be credited for not simply inferring differences in effects from differences in separate probability values (Krivo and
Peterson, 2000; Lee, 2000; Phillips, 2002). Interestingly, these three studies came to
very different conclusions regarding the impact of race-specific economic deprivation
on race-specific homicide rates. Krivo and PetersonÕs (2000) analysis was based on
1990 city-level data and used an innovative race-specific measure of deprivation designed to tap the concept of concentrated disadvantage. Consistent with OuseyÕs
study (1999) study, Krivo and Peterson concluded that the relationship between concentrated disadvantage and the log transformed homicide rate is significantly stronger for whites than it is for blacks (b = .323, SE = .078 versus b = .171, SE = .063, a
2
Sampson (1987, p. 372) points out that the known census undercount in black communities could affect
the ‘‘measurement properties’’ of various race-specific demographic variables and therefore the precision
of estimates for race-specific homicide rates. Moreover, while researchers seem to have settled on a limit of
at least 5000 blacks for their urban samples to ensure reliable estimation, this limit is arbitrary.
3
It is also not uncommon to have a non-significant coefficient that is larger than its statistically
significant counterpart.
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difference that was apparently significant at the one-tailed p < .01 level). LeeÕs (2000)
race-specific analysis was also based on 1990 city data and focused on the relationship between concentrated poverty and log transformed homicide rates. Similar to
Krivo and Peterson (2000), Lee reported that the estimate for concentrated poverty
for whites (b = 3.321, SE = 1.404) was larger than that for blacks (b = 2.416,
SE = 1.107), but unlike Krivo and Peterson, LeeÕs test for coefficient differences suggested this apparent discrepancy was not statistically significant. Phillips (2002) conducted a race-specific analysis of cross-sectional variation in square root transformed
homicide rates using 1990 metropolitan area data. Inconsistent with Ousey (1999),
Lee (2000), and Krivo and Peterson (2000), Phillips reported that the relationship
between various poverty-related variables and the square root of the homicide rate
appeared marginally stronger for blacks and Latinos than for Non-Hispanic whites.4
While all of these recent studies have greatly contributed to the literature both in
terms of technical and conceptual innovations, following a common practice in dozens of studies before them, they have also overlooked an important methodological
concern. We argue that the result of this methodological oversight is far from trivial
for answering the important theoretical questions that motivated both contemporary
and classic studies in this area.5 Moreover, we suggest that like the recent normative
shift in this literature toward testing coefficient differences rather than simply comparing p values, closer attention to the interpretation of models with transformed
variables in future studies will significantly advance the scientific investigation of racial differences in crime rates.
3. Variable transformations and interpreting effects in race-specific homicide studies:
the logarithm of the homicide rate is not the homicide rate
The most commonly stated reason for applying a transformation to the dependent
variable is to reduce significant positive skew. The use of a skewed-dependent variable increases the likelihood of influential outliers and non-constant variance, and
obfuscates the usual interpretation of the coefficients in terms of central tendency.
4
Phillips speculated that there were three potential reasons why her findings differed so much from those
of previous studies. First, her study compared Non-Hispanic whites to Non-Hispanic blacks and Latinos,
while most studies did not distinguish racial groups by ethnicity. Second, Phillips used metropolitan
statistical areas as the primary unit of analysis, while most others used cities. Third, while almost all
previous studies used the log transformation to reduce skew, she used the square root transformation (a
much milder adjustment, see Tukey, 1954).
5
These studies are certainly not the only ones with this oversight, and thus our intention is not simply to
replicate, comment on, or critique these four works. Indeed, we know of many other published analyses
with misinterpreted semi-log coefficients, including some by the authors of the present paper. We argue
that the lack of appropriate attention to data transformations is so widespread in criminology (especially
macro-criminology) that it has become normative in the discipline and only offer these four studies as
prominent recent examples focused on the same substantive issue: racial comparison of effects. Moreover,
issues concerning transformation and re-transformation are most clearly evident in these recent studies
because they have gone significantly beyond earlier research in this area by correctly insisting on testing for
significant differences in effects between racial groups.
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The logarithmic and square root transformations are two common functions used to
enhance the accuracy and efficiency of least squares estimation by reducing positive
skew in the dependent variable. The logarithmic transformation is by far the more
popular of the two and is also more powerful (i.e., more effective for cases of severe
positive skew).6
The beauty of the log transformation is that logarithmic metric results can often
be quickly interpreted for their relevance to the original metric. As Manning (1998,
p. 285) points out in his study of the use of the log transformation for estimation in
health economics,
Although such estimates may be more precise and robust, no one is interested
in log model results on the log scale per se. Congress does not appropriate log
dollars. First Bank will not cash a check for log dollars. Instead, the log scale
results must be retransformed to the original scale so that one can comment on
the average or total response to a covariate x.
In terms of macro-level criminological research, politicians do not pledge to reduce
the log of the crime rate and the major theories are not designed to explain variation
in the logarithm of homicide incidence. A rather extensive literature has developed in
other disciplines around the most accurate retransformation method, so that the log
transformation can be used to estimate precise and robust effects that can then be
interpreted for relevance to the theoretically meaningful outcome variable (Duan,
1983; Thornton and Innes, 1989; Van Garderen and Shah, 2002). Unfortunately,
macro-level studies in criminology have mostly ignored issues of variable retransformation and even the basics of the approximate interpretation of log model coefficients are rarely evident.
OsgoodÕs (2000) important study on Poisson-based regression is an exception.
Describing the approximate retransformation of coefficients in a semi-log model,
Osgood (2000, p. 34) notes
Under a linear model of the untransformed data, the regression coefficients indicate the difference in the mean of the dependent variable that is associated with a
unit difference on the explanatory variable. After the logarithmic transformation, the regression coefficients reflect proportional differences in the mean of
the dependent variable, given a one-unit difference on the explanatory variable.7
Although several data-specific factors can affect the mathematical accuracy of this
interpretation, it is still sufficient for many purposes and much better than simply
6
In fact, it is quite rare to find a race-specific study that does not logarithmically transform the
dependent variable (e.g., Smith, 1992). Moreover, as Land et al. (1990) suggest, the log transformation is
so popular that an appropriate rationale for its use often goes unstated.
7
Osgood (2000, p. 34) rightfully describes this interpretation as ‘‘conceptual.’’ As noted earlier, entire
studies have been devoted to delineating the mathematically most precise interpretation of semi-log
estimates in terms of the original metric (Duan, 1983; Ai and Norton, 2000; Manning, 1998; Manning and
Mullahy, 2001).
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interpreting the coefficients in the same way that one would for the untransformed
data (i.e., ignoring differences between additive effects and proportional effects and
implicitly assuming that the crime rate and the log of the crime rate are essentially
the same thing).
Arguably, this basic interpretation of semi-log coefficients is less relevant for those
researchers who simply want to see whether a first-order effect is statistically significant in a specific sample. However, the conceptual interpretation of the semi-log
coefficients as representing proportional change from the mean of Y in the original
metric is extremely important for those interested in comparing effects across groups.
Simply put, a small proportional change from a big mean can be equivalent to a big
proportional change from a small mean. Recognizing this is crucial for distinguishing between significant differences in effects and just significant differences in means.
To illustrate this point more clearly, Table 1 presents artificial data for two samples with uniform distributions, both having perfectly linear relationships between
the X and Y variables (representing poverty and homicide rates, respectively) and
the same additive difference between each of the points. What distinguishes the
two samples is their significantly different means: 12 for the low poverty/low homiTable 1
An illustration of the range-specific nature of semi-log coefficients
Group 1: Low poverty/low homicide
Group 2: High poverty/high homicide
Panel A: Level model
Poverty
X
10
11
12
13
14
Panel C: Level model
Poverty
X
30
31
32
33
34
Homicide
Y
10
11
12
13
14
Level equation
Y = 1x
Level equation
Y = 1x
Additive estimate
b=1
Additive estimate
b=1
Panel B: Semi-log model
Poverty
X
10
11
12
13
14
Log homicide
Y (ln)
2.303
2.398
2.485
2.565
2.639
Panel D: Semi-log model
Poverty
X
30
31
32
33
34
Semi-log equation
lnY = 1.470 + .0840x
Semi-log equation
lnY = 2.464 + .0313x
Proportional estimate
b = .0840
Proportional estimate
b = .0313
Homicide
Y
30
31
32
33
34
Log homicide
Y (ln)
3.401
3.434
3.466
3.497
3.526
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cide sample versus 32 for the high poverty/high homicide sample. Panels A and C
illustrate that regressing the untransformed Y variables on their respective untransformed X variables would lead one to conclude that the effects are identical for both
samples (b = 1). Panels B and D demonstrate that regressing the natural log of the Y
variables on their respective untransformed X variables might now lead one to conclude that the effects are qualitatively different for the two samples (b = .0840,
SE = 0.0024 versus b = .0313, SE = .0003, p < .001 for the coefficient difference).
However, recalling OsgoodÕs (2000, p. 34) approximate interpretation of these coefficients as ‘‘proportional differences in the mean of the dependent variable, given a
one-unit difference on the explanatory variable,’’ these results are not at all surprising since again, taking a large proportion of a small mean can be equivalent to taking
a small proportion of a large mean (i.e., .0840 of 12 is basically equivalent to .0313 of
32; both are approximately 1). While data in the real world are almost always more
complicated, this simplified example illustrates the essential problem of comparing
semi-log coefficients directly across models for different samples.8 Apparent differences in effects may simply reflect differences in means. Thus, it is possible that the
substantially smaller semi-log coefficients for blacks reported in some previous studies may simply indicate that blacks generally have substantially higher levels of structural disadvantage and homicide than whites.
The present paper demonstrates a few research strategies for overcoming this
methodological obstacle. Specifically, we note that linear models with robust standard errors and double-log models with both the dependent and independent variables log transformed can facilitate an intuitive comparison of race-specific
coefficients. Semi-log models can also be used to produce a meaningful comparison
of race-specific effects when used in conjunction with a set of calculations to compute
the retransformed coefficients and their associated confidence intervals.
4. Data, variables, and method
Similar to previous studies, we analyzed 1990 data for a sample of 134 cities with
total populations greater than 100,000 and a black population of at least 5000 in order to determine whether the effect of poverty on homicide varies by race. While 154
8
As Manning (1998, p. 290) has noted in his analysis of the problems affecting the interpretation of
coefficients in semi-log models,
In the case of untransformed dependent variables, the mean response can be estimated by using the
mean x multiplied by an unbiased estimate of b. This ability is one of the convenient properties of
the OLS estimate that disappears when the dependent variable is transformed. . . Instead, the mean
response is the mean of the retransformed estimate of y, which depends on the distribution of the
xÕs not just their mean.
Further complicating matters is the frequent need to include a polynomial term to account for nonlinearity created by the log transformation of the dependent variable (Hannon and Knapp, 2003; Nelson,
1981). This complication is most likely to arise in cases where there is a wide range of values for the
untransformed dependent variable.
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cities initially met this criterion, several cities (particularly those in Florida) were excluded because they lacked complete data for homicide rates, while several others
lacked data on segregation. One city, Oxnard, California, was omitted from the analyses because an examination of residuals and CookÕs D statistics suggested that it
was an excessively influential outlier in the models for whites.
Also similar to previous studies, we used data originally derived from the Comparative Homicide File (CHF) which is based on the FBIÕs annual Supplementary
Homicide Reports. Race-specific rates of homicide offending for single-offender/single-victim homicide incidents were calculated in the usual fashion: the number of
homicides involving an offender of a given race divided by the total race-specific population, multiplied by 100,000. Consistent with earlier research, an imputation algorithm developed by Williams and Flewelling (1987) was used to extrapolate data to
cases where the offenderÕs race was unknown and race-specific rates were averaged
over five years (1987–1991) to minimize the impact of random fluctuations (see Parker and Pruitt, 2000b for a list of adjusted CHF race-specific city homicide rates).
Nineteen-ninety census data were used to create the independent variables. These
included race-specific measures of the poverty rate, the percentage of owner-occupied housing units, the percentage of the population that is male and between the
ages of 15 and 34, and the percentage of the population that is of Hispanic origin.
Additionally, we controlled for several city-wide factors: the index of dissimilarity
(originally calculated based on tract data for black and white residents), logarithmically transformed population size, and dummy variables for regional location in the
South and West.
We focus on race-specific poverty rates rather than employing a combined index
of disadvantage as some previous studies have done because there seems to be widespread agreement that the poverty rate reflects economic deprivation, while other
measures often included in disadvantage indices have been interpreted in various
ways (e.g., the prevalence of female-headed families as reflecting resource deprivation in one view versus reflecting improper socialization and monitoring of male
youth in another). Moreover, there is little agreement on which factors should be included in the disadvantage index (e.g., some include segregation while others do not).
Thus, in order to ensure clarity in the presentation of both our substantive and methodological arguments, we employ a relatively parsimonious model and leave more
complex model specifications for future research.
We incorporate the race-specific percentage of owner-occupied housing units in
our analyses because of its demonstrated relevance in previous studies (e.g., Krivo
and Peterson, 2000; McNulty, 2001) and because of its theoretical significance as a
measure of residential stability. We include the race-specific percentage of young
males for consistency with previous studies and because they are the primary demographic group involved in homicide offending. While only a few studies have controlled for the percentage of the population of Hispanic origin, we agree with
Nelsen et al. (1994) and Parker and McCall (1999) that it is an important control
variable since it helps adjust for the limitations of race-specific official homicide statistics. The index of dissimilarity, a commonly used measure of residential segregation, is included in the analyses to examine the potentially unique criminogenic
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influence of community social isolation (Massey and Denton, 1993). The controls for
population size and region (both South and West) were included primarily to facilitate comparison with previous research.
Following Ousey (1999), we employ seemingly unrelated regression models to account for the fact that the two samples being compared are not independent; both
are based on data for the exact same cities. Also following Ousey, we use an
across-equation F test to test for statistically significant differences between effects.
We estimate three separate sets of equations to assess whether the effect of poverty
on homicide is racially invariant and to demonstrate how a direct comparison of
coefficients across semi-log models can be extremely misleading. The first set shows
the race-specific effects from level (linear/untransformed) models using WhiteÕs
(1980) robust standard errors. The second set presents the race-specific coefficients
from semi-log models, with the dependent variables logarithmically transformed.
The third set illustrates the effects from double-log models, with both the dependent
and independent variables logarithmically transformed (except the two regional
indicators).
5. Results
5.1. The level, semi-log, and double-log models
Table 2 presents the means and standard deviations for the variables in the black
and white homicide offending rate equations. The univariate statistics reveal some
substantial differences between means for the two groups. Similar to means reported
in previous studies (e.g., Ousey, 1999), the average black homicide offending rate is
nearly five times the white rate (40.99 and 8.76, respectively). Furthermore, the mean
poverty rate is nearly three times higher among blacks than whites, while the mean
percentage of the population of Hispanic origin is nearly four times higher among
whites than blacks.
Table 2
Means and standard deviations of variables by racial group
Variable
Untransformed homicide rate
Homicide rate (ln)
Poverty rate
Residential segregation (D)
Percent owner occupied housing
Percent Hispanic origin
Percent males age 15–34
Population size (ln)
West
South
Black population
White population
Mean
SD
Mean
SD
40.99
3.58
28.99
58.05
36.33
2.26
17.16
12.45
0.24
0.38
19.97
0.55
8.28
14.51
9.66
2.57
2.52
0.77
0.43
0.49
8.76
1.91
11.09
58.05
55.95
8.83
17.62
12.45
0.24
0.38
6.56
0.75
3.97
14.51
8.92
11.80
2.44
0.77
0.43
0.49
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Table 3 displays the results of the multivariate seemingly unrelated regression
(SUR) analyses for white and black homicide offending rates. As mentioned previously, SUR is used rather than ordinary least squares (OLS) because the two samples
are not independent, which means the disturbance terms for the models will tend to
be correlated. We also examined the models for excessive multicollinearity and found
that the variance inflation factors (VIFs) were all below four for both the white and
black models.
Table 3, Panel A presents results for the level models for blacks and whites
using the untransformed homicide rates (and robust standard errors). Six of the
eight independent variables had statistically significant effects in the white model
(the poverty rate, percent Hispanic origin, percent males 15–34, log population
size, and West and South), while five of the variables exhibited statistically signif-
Table 3
SUR parameter estimates for black and white homicide offending rates (N = 134 cities)
Black homicide
White homicide
Poverty F test
Panel A: Level
Poverty rate
Segregation (D)
% Owner occupancy
% Hispanic origin
% Males age 15–34
Population size (ln)
West
South
Constant
0.443
0.395
0.139
0.935
1.065
5.667
20.674
6.989
61.802
(0.210)*
(0.137)*
(0.189)
(0.648)
(0.780)
(2.083)*
(4.041)*
(3.506)*
0.351
0.027
0.036
0.217
0.585
2.206
3.119
2.406
15.430
(0.096)*
(0.036)
(0.062)
(0.042)*
(0.179)*
(0.634)*
(1.296)*
(0.785)*
0.247
Panel B: Semi-log
Poverty rate
Segregation (D)
% Owner occupancy
% Hispanic origin
% Males age 15–34
Population size (ln)
West
South
Constant
0.018
0.011
0.004
0.015
0.020
0.132
0.541
0.132
0.719
(0.006)*
(0.004)*
(0.005)
(0.017)
(0.021)
(0.064)*
(0.124)*
(0.102)
0.050
0.007
0.005
0.021
0.077
0.227
0.545
0.372
0.699
(0.013)*
(0.005)
(0.006)
(0.004)*
(0.020)*
(0.073)*
(0.144)*
(0.115)*
6.324*
Panel C: Double-log
Poverty rate (ln)
Segregation (D) (ln)
% Owner occupancy (ln)
% Hispanic origin (ln)
% Males age 15–34 (ln)
Population size (ln)
West
South
Constant
0.530
0.678
0.250
0.123
0.174
0.114
0.508
0.159
2.991
(0.146)*
(0.206)*
(0.147)
(0.064)
(0.382)
(0.060)
(0.123)*
(0.102)
0.664
0.347
0.012
0.238
1.298
0.202
0.461
0.449
0.509
(0.121)*
(0.229)
(0.322)
(0.042)*
(0.367)*
(0.069)*
(0.139)*
(0.111)*
0.667
Note. Standard errors in parentheses.
*
p < .05 (two-tailed).
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13
icant relationships in the black model (the poverty rate, residential segregation, log
population size, and West and South). In terms of the effect of the poverty rate, the
primary focus of the current paper, the estimate actually appeared slightly larger in
the black model than the white model, with the results suggesting that a one-unit
increase in the poverty rate would increase the homicide rate by .443 for blacks
and .351 for whites. However, the F test for equality of effects indicated that this
difference is clearly not statistically significant. Thus, in the level models there is no
solid evidence supporting the notion of a differential effect of poverty for the two
groups.
Panel B presents results for the more commonly used semi-log models, where
the homicide rates have been logarithmically transformed to reduce positive skew.
The coefficients for these models are very similar to those reported in Parker and
PruittÕs (2000a) study, which also focused on the effect of poverty on race-specific
homicide. Specifically, the poverty rate was a significant predictor for both blacks
and whites, but the effect on the logged homicide rate appeared roughly three times
greater in the white model than in the black model. The coefficients suggest that a
one-unit increase in the poverty rate would, on average, increase the logarithm of
the black homicide rate by .018, while the same change in poverty would increase
the logarithm of the white homicide rate by .050. The F test for the significance of
the difference between the estimates revealed that this discrepancy was indeed statistically significant (p < .05). However, as noted earlier, showing a significant difference in povertyÕs effect on the logarithm of the homicide rate is not the same
thing as showing a significant difference in povertyÕs effect on the homicide rate.
The proportional effects implied by the semi-log model are multiplicative and
inherently range-specific. This means that, unlike the level (linear/untransformed)
model, the meaningfulness of the average effects will be determined, in part, by
the mean of the dependent variable in the original metric. Thus, the significantly
lower semi-log coefficient for blacks may simply reflect their well-documented significantly higher mean homicide rate.
Panel C shows the results from the double-log models, where both the dependent variable and explanatory variables are logarithmically transformed. Like the
semi-log model, the slopes in the double-log model represent expected relative or
proportional change in the original y, but unlike the semi-log model this relative
change is not measured in terms of absolute change in x, but rather proportional
change in x. Thus, the issue of lesser proportions of larger values being equivalent
to greater proportions of smaller values is roughly balanced out by the double
transformation, since both the dependent variable and the explanatory variables
are similarly expressed (and similarly constrained). In the double-log model, the
estimated coefficients indicate the percent change in the dependent variable associated with a one-percent change in the independent variable. The poverty coefficient
for whites (b = 0.664) suggests that a 1% increase in the poverty rate would increase the homicide rate by 0.664%, while the coefficient for blacks (b = 0.530) suggests that a 1% increase in the poverty rate would increase the homicide rate by
0.530%. Like the results from the level models, the difference in the estimated coefficients is not statistically significant. Thus, there is no reliable evidence to support
ARTICLE IN PRESS
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L. Hannon et al. / Social Science Research xxx (2005) xxx–xxx
the claim that the effect of poverty on homicide differs by race in the double-log
models.9
In sum, the level and double-log results suggested that the apparent racial differences in the estimated effects of poverty were small and not statistically significant
(an estimated 26% stronger for blacks in the level models and 25% stronger for
whites in the double-log models). In contrast, the apparent racial difference in the
poverty coefficients was substantial and statistically significant in the semi-log models (an estimated 278% stronger for whites). However, interpreting the semi-log coefficients for their relevance to the homicide rate (not the log of the homicide rate), the
results for Panel B may ultimately suggest the same conclusion as those in Panels A
and C: the effect of poverty on the homicide rate is not significantly different for
whites and blacks.
5.2. Retransforming semi-log estimates for group comparisons
5.2.1. Calculating retransformed coefficients
As noted earlier, the issue of retransformation of semi-log coefficients has rarely
received attention from criminologists. When it has been addressed (e.g., Hannon
and Knapp, 2003; Osgood, 2000), it has usually only been in conceptual terms. In
contrast, researchers in other disciplines have devoted a considerable amount of
attention to solving problems affecting the mathematical precision of various
retransformation procedures (e.g., Duan, 1983; Manning, 1998; Van Garderen and
Shah, 2002). Drawing from this literature, this section provides a mathematically
precise retransformation formula.10 This procedure has the advantages of producing
accurate retransformed coefficients and of making explicit several decisions that
researchers must face when implementing a retransformation. Moreover, the procedure we outline allows for alternative assumptions about the errors and can be used
with Poisson-based models.
To begin, the estimated semi-log equation takes the form
^ þ ei
ln Y i ¼ ^
a þ x0i b
ð1Þ
for each cross-section unit i (here, cities); xi is a vector of independent variable values
for unit i, a and b refer to model parameters; Ù indicates estimated values; and e is a
random error.
9
In one set of alternative analyses we found that replacing the log transformation with the square root
in the double transformation models produced practically identical race-specific poverty coefficients.
Moreover, even the models with the untransformed poverty and square root transformed homicide rates
produced race-specific coefficients that were not significantly different from one another, illustrating the
difference in power between the logarithmic and square root functions (see Tukey, 1954; for a hierarchy of
transforms). We also investigated the possibility that a quadratic term for poverty needed to be included in
the various equations to account for significant non-linearity in its relationship to homicide (or the log of
homicide). The coefficients for the quadratic terms were not statistically significant at the .05 level in any of
the equations. In yet another set of alternative analyses, we replicated the basic findings in Table 3 using
1999–2001 race-specific homicide arrest data and demographic variables from the 2000 Census.
10
The following discussion draws from Manning (1998) and Ai and Norton (2000). These articles
provide a more detailed exposition of the exact retransformation procedure described here.
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15
Retransformation requires four steps. First, take the anti-log of both sides of Eq.
(1) to obtain an expression for the level of Yi:
^ þ ei Þ;
a þ x0 b
ð2Þ
Y i ¼ expð^
i
or
^ expðei Þ.
a þ x0i bÞ
Y i ¼ expð^
Second, take the expected value of Yi given xi, denoted E (Yi|xi)
^
a þ x0 bÞEðexpðe
EðY i jxi Þ ¼ expð^
i ÞÞ.
i
ð2aÞ
ð3Þ
Third, obtain the response of E (Yi|xi) to a change in the kth variable of interest (e.g.,
poverty), xik, by differentiating E (Yi|xi) with respect to xik11
oEðY i jxi Þ
^
¼ bk expð^
a þ x0i bÞEðexpðe
ð4Þ
i ÞÞ.
oxik
Finally, compute the average response across all cross-section units by summing Eq.
(4) over i and dividing by the number of units, N (here, 134)
N
oEðY jxÞ 1 X
^ expð^
^
b
¼
a þ x0i bÞEðexpðe
i ÞÞ.
oxk
N i¼1 k
ð5Þ
In general, Eq. (5) can be easily computed given values for xi and given a value for
E (exp (ei)).12 How one obtains an estimate of E (exp (ei)) requires some elaboration.
Because the ei are the errors from a regression, their mean is zero. However, because
the anti-log is a non-linear function, it does not follow that the mean of exp (ei) is 1.
Rather, its particular value depends on the specific distribution assumed for the ei. A
common assumption is that the ei are distributed normally, with mean 0 and variance
re. In this case, E (exp (ei)) equals exp (0.5re) (Ai and Norton, 2000; Manning, 1998).
One can then use the sample variance of the ei to estimate re, and plug in the result to
calculate a magnitude for Eq. (5).
Making a specific assumption about the distribution of the ei can be problematic
either because one does not know the underlying distribution or because the sample
sizes are small. For instance, the small sample distribution of the ei might not closely
approximate a normal distribution, and so a value of exp (0.5re) might produce a
very biased estimate of E (exp (ei)). An alternative non-parametric approach, which
is preferred when samples are small, is to estimate E (exp (ei)) using a ‘‘smearing factor’’ (Ai and Norton, 2000; Duan, 1983). The smearing factor, D, is computed simply
as the sample average of exp (ei)
D¼
N
1 X
expðei Þ.
N i¼1
ð6Þ
11
The calculation assumes the ei are not a function of the xi (see Manning (1998) and Ai and Norton
(2000) for the alternative case).
12
It is important to note that Eq. (5) indicates that the average response of Y to a change in a specific
variable is a function of all the coefficients and variable values in the model. Also, because for a non-linear
function, a function of the mean does not equal the mean of the function, Eq. (5) requires summing Eq. (4)
over each cross-section unit and then dividing by N, and not simply using sample mean values.
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Substituting Eq. (6) into Eq. (5) yields a generally applicable retransformation
expression, one especially suited to small sample estimation
N
oEðY jxÞ 1 X
^
¼
b expð^
a þ x0i bÞD.
ð5aÞ
oxk
N i¼1 k
At times, researchers will want to compare retransformed coefficients from different semi-log or Poisson/negative binomial equations. A case in point is our comparison of the response of homicide rates to poverty rates for whites and blacks. Here,
there will be two estimates of Eq. (5a)—one based on the estimated coefficients and
smearing factor for the white data, and one based on the estimated coefficients and
smearing factor for the black data. When such a comparison is desired, one must decide on the appropriate values for xi to use for evaluating (5a).
Because Eq. (5a) is non-linear, the size of the retransformed coefficient will be
range-specific; in other words, it will depend on the specific values of xi used. A
straightforward approach is to simply use the actual black and white xi values for
each of the 134 cities and then compare the resulting average responses for blacks
and whites. Emphasizing the central tendency of responses observed in the actual
data is logically consistent with the standard presentation of regression results. Other
choices for the comparison could be made, but they should be theoretically motivated and explicitly justified.
Applying Eq. (5a) to our race-specific data and using the black and white semi-log
SUR parameter estimates from Table 3 (Panel B) produced a retransformed average
estimate for povertyÕs effect on the black homicide rate that actually appeared larger
than the retransformed average poverty estimate for whites (.442 for whites versus
.744 for blacks). Interestingly, in this particular case, the estimates using the more
precise retransformation procedure and smearing factor recommended by Manning
(1998) were very close to those that would have been obtained following OsgoodÕs
(2000) approximate retransformation procedure (calculated simply as the mean
untransformed homicide rate multiplied by the semi-log coefficient: 8.76 * .050 =
.438 for whites versus 40.99 * .018 = .738 for blacks).
The results of both of these retransformation procedures clearly contradict the
common incorrect interpretation of the semi-log results presented in Table 3 (Panel
B) as indicating a significantly stronger average effect of poverty on the white homicide rate. Indeed, the only question remaining at this point is if the apparent higher
average effect of poverty for blacks is significantly higher. Testing this new hypothesis that the average effect of poverty on the homicide rate is actually significantly
stronger for blacks than whites requires a formal test of the significance of the difference in coefficients.
5.2.2. Testing for a significant difference in retransformed coefficients
Once the retransformed coefficient values are obtained for each racial group,
researchers will often want to know whether any measured difference is statistically significant. Earlier in this paper, we argued that homicide studies comparing effects across
groups should generally formally test for the significance of the difference. A formal test
requires knowledge of the confidence interval of the difference in group coefficients.
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As seen in Eq. (5a), the retransformed coefficients are complex non-linear functions of all the estimated coefficients and the smearing factor, each of which has
its own distribution and standard error. In certain cases, one can appeal to asymptotic distribution theory to analytically derive the standard errors of the retransformed coefficients. Once again, however, asymptotic theory can be a very poor
guide when samples are relatively small.13
An alternative approach, and one that has been increasingly used in other disciplines, is to derive the needed confidence interval numerically via bootstrap simulations (MacKinnon, 2002). Bootstrapping uses the estimated errors from Eq. (1)
to generate a distribution of estimated values for Eq. (5a). The essential idea is
that the estimated ei from Eq. (1) describe the distribution of shocks to ln (Yi)
for each i cross-sectional unit. However, the observed way in which the ei hit each
ln (Yi) is only one possibility. It is possible, for example, that e1 would hit ln(Y1)
in the observed sample, but that e1 would hit ln (Y2) in another sample. If so,
then each set of xi values would be associated with different values of ln (Yi) than
before, and so different estimates of a and b would result. If enough different
combinations of the ei are considered, then one can generate a whole distribution
of values for Eq. (5a). This distribution, in turn, permits construction of the confidence interval.
Jeong and Maddala (1993) describe how to implement the bootstrap technique:
1. Compute predicted residuals for each of the n cross-section units
^
ei ¼ ln Y i ð^
a þ x0i bÞ.
2. Resample ei: obtain ei by drawing n times at random with replacement from ei.
3. Construct a Ôfake dataÕ ln (Yi)* by the formula
^ þ e .
lnðY i Þ ¼ ^
a þ x0i b
i
4. Use the data x and ln (Yi)* to re-estimate the regression coefficients.
5. Re-compute the value of Eq. (5a).
6. Replicate steps 2 through 5 m times, where m is a large number (here, 10,000).
The results of the bootstrap simulations can then be used to obtain the needed
confidence interval. We consider two alternatives: a bias-corrected percentile confidence band and a bias-corrected bootstrap interval (Jeong and Maddala, 1993;
MacKinnon, 2002). We compute these for the difference between the estimated
retransformed poverty coefficients for blacks and whites. The bootstrap simulations produced a sample distribution of 10,000 values indicating differences between retransformed poverty coefficients, with a mean difference between
coefficients of .298 (white estimate minus black estimate). To construct a percen13
Ai and Norton (2000) derive analytical representations of the asymptotic standard errors for
retransformed coefficients under various assumptions about the regressions errors. They do so only for
single-equation estimates, not for system estimates such as SUR. They also note the dangers of using the
asymptotic formulas for small samples, and recommend bootstrap techniques as an alternative method.
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L. Hannon et al. / Social Science Research xxx (2005) xxx–xxx
tile confidence band, one simply orders the estimated poverty coefficient differences from smallest to largest, and then identifies the values at the 2.5 and 97.5
percentiles (the 250th and 9750th values). These values demarcate the 95% confidence interval. After a very slight adjustment for bias (MacKinnon, 2002, p. 639),
the resulting 95% confidence interval was .738 to .111. As always, since the 95%
confidence interval around the average estimate contains zero, the null hypothesis
that the estimate is different from zero cannot be rejected at that level. The alternative calculation, a bias-corrected bootstrap interval, uses the standard error of
the bootstrap coefficient estimates and the desired t values to construct a 95% confidence band in the standard way (i.e., plus or minus 1.96 times the bootstrap
standard error). In this case, the bootstrap confidence interval was virtually the
same as the percentile interval. Once again, the null hypothesis could not be rejected at the p < .05 level.
Addressing a somewhat unexpected question given the existing literature on this
topic, we conclude from these calculations that the average effect of poverty for
blacks is not significantly greater than that of whites. Ultimately then, like the results
for the double-log and level analyses (Table 3), a close look at the semi-log model
results actually suggests that there is no discernible difference for blacks and whites
in the effect of poverty on the homicide rate.
6. Conclusions
Macro-criminologists have devoted a considerable amount of attention to the issue of whether the relationship between poverty and violent crime is racially invariant. For example, in line with traditional social disorganization theory, Sampson
and Wilson (1995, p. 41) have argued that the ‘‘sources of violent crime appear to
be remarkably invariant across race and rooted instead in structural differences
among communities, cities, and states in economic and family organization.’’ Arguing for the necessary integration of insights derived from more culturally oriented
perspectives, others have suggested that poverty may have a relatively weak effect
on black rates of violence because of historically developed ‘‘cultural and normative
adaptations’’ that eclipse structural conditions as the primary source of violent crime
(Ousey, 1999, p. 421).
The empirical literature on this important substantive issue has been hampered by
two crucial methodological problems. First, many studies have erroneously concluded from across-sample differences in p values that the actual effects differ between racial groups. Second, the relatively few studies formally testing for effect
differences have unfortunately followed a tradition in macro-criminological research
of interpreting racial differences in semi-log coefficients in the same way that one
would if the dependent variables were not logarithmically transformed, implicitly
suggesting that showing differences for povertyÕs effect on the logarithm of the homicide rate is the same thing as showing differences for povertyÕs effect on the homicide
rate. In the present paper, we demonstrate that the two can be very different. In fact,
our analyses suggest that substantive inferences can be completely opposite depend-
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19
ing on whether a researcher compares effects in the logarithmic or original metric
(e.g., what appears as a 300% stronger effect for whites in the log metric may actually
translate into a 100% stronger effect for blacks in the original metric).
The beauty of the log transformation is that it can be used to avoid the estimation problems that arise when one uses the untransformed dependent variable,
while at the same time producing results that can be discussed in terms of the original metric, provided that one does some additional calculations. These additional calculations are needed because the average reader does not have
‘‘logarithmic eyes’’ (Myers, 1947, p. 17), and because the average reader operates
under the assumption of a ‘‘general linear reality’’ (Abbott, 1988). Right or wrong,
most people assume that social relationships follow a linear pattern, and thus in
the absence of retransformation calculations it becomes crucial for researchers
to explicitly inform the reader when, why, and what type of ‘‘non-linearity’’ is
assumed.
Including these additional retransformation calculations in our analysis of the
relationship between race-specific poverty and homicide rates, we find no evidence
of a weaker relationship for blacks than whites. Furthermore, easily implemented
alternative methodological approaches such as race-specific double-log models and
level models with robust standard errors led to the same conclusion.14 Consistent
with social disorganization theory, the relationship between poverty and homicide
appears to be quite similar for blacks and whites.
Of course, it is not at all clear whether the apparent invariance in povertyÕs effect
on race-specific homicide reported in this study will mean that other measures of disadvantage will be invariant in their effects for blacks and whites (or that other model
specifications and units of analysis will produce identical results). Future research
should investigate potential race differences in the effects of other types of disadvantage, preferably using both cross-sectional and longitudinal research designs. Moreover, future analyses should continue to investigate similarities and differences in
povertyÕs effect on violence in predominately black and white neighborhoods, since
aggregation to the city level may obscure differences in the processes generating violent crime. What is absolutely clear, however, is that future studies comparing effects
across groups need to take account of their variable transformations (or implicit
transformations as in Poisson-based models15). Without attention to the retransfor14
The double-log model is generally an efficient option, since it utilizes the log transformation to mitigate
the estimation problems associated with positive skew and also produces coefficients which can be directly
compared across race-specific samples. Given recent advances in the functionality of statistical software
packages, we also recommend the level/linear model using robust regression (e.g., the proc robustreg
command in SAS and the rreg command in Stata) as an easily implemented alternative to the semi-log
model with retransformation calculations.
15
As Osgood (2000, p. 24) notes, the role of the natural logarithm in the basic Poisson regression is
‘‘comparable to the logarithmic transformation of the dependent variable that is common in analysis of
aggregate crime rates.’’ Thus, standard Poisson-based models produce results in the logarithmic metric
(i.e., they model the log of an event count, or alternatively, the log of an event rate if a population offset is
used). Of course, one important advantage of Poisson-based models over semi-log equations is that the
implicit log transformation in Poisson-based models does not require adding an arbitrary constant to
accommodate zero values in the original metric.
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mation of coefficients expressed in the logarithmic metric, researchers in this area
(and others) will continue to report precisely estimated, robust, and dramatic group
differences, with very little real substantive meaning.
Acknowledgments
We thank Steven Messner, Kenneth Land, and the anonymous reviewers for their
helpful comments. This article also benefited from thoughtful questions by Matt Lee
and Graham Ousey in response to an earlier paper presented at the 2003 Southern
Sociological Society meetings in New Orleans. In addition, we gained useful insights
about the mechanics of Poisson-based models and the logarithmic funciton from
Steve SimonÕs comprehensive statistics webpage (www.cmh.edu/stats/). All errors
are, of course, ours alone.
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